all side stitches have been worked. End with a purl row.
I have 15 stitches on my needle. I'm okay with rows 1-4 the first time through. The instruction to Repeat rows 3-4 increasing 1 additional K or P until all side stitches have been worked is throwing me. After the second pass through row 4, I have no more purl stitches to p2tog, p1. Am I supposed to add purl stitches to finish out the row??? I'm also confused as to what is meant by a side stitch? Are these the stitches that are left behind when I turn the work?

Yes, the 'side' sts would be the ones you left unworked when you started doing the short rows. You would take a stitch from them to do the p2tog, I think.

I understand but it seems that I'm running out of stitches on the purl side. In other words, I can do row 3 (knit) just fine. When I turn to do row 4 (purl) there are not enough stitches to complete the instructions on row 4. The first time I knit/purl rows 3 and 4, all is well, it is the instruction that has me repeat rows 3 and 4 has me confloozled when I don't have enough stitches to do row 4. Am I supposed to add stitches to make it work? What does the author mean by "Repeat rows 3 and 4, increasing 1 additional k or p stitch until

You don't need to add any more sts onto the needles. Just use one additional st from the ones that are waiting on the sides. There is a mistake in the pattern, so no wonder you had trouble.
"row 1: k8, k2 tog, k1, turn
row 2: p3, p2 tog, p1, turn
row 3: k4, k2 tog, k1, turn
row 4: p5, p2 tog, p1, turn
Repeat rows 3 and 4, increasing 1 additional k or p stitch until
all side stitches have been worked. End with a purl row."
In the repeats of rows 3 and 4 you need to work to one st before the gap. So in repeat of row 3, it's k6 and in repeat of row 4, it's p7. That uses all the sts-in-waiting and ends on a purl row so that your ready for the next instruction on a knit row.

The numbers in the brackets are the number of stitches for the size, which should be small, med, large, or a number size like 10, 12, 14. I love to knit socks and use the eye of partridge pattern for the heel flap